Prove that \[ \frac{\sin x+\sin 2x}{1+\cos x+\cos 2x}=\tan x \]
Proof:
\[
LHS=\frac{\sin x+\sin 2x}{1+\cos x+\cos 2x}
\]
Using the identity:
\[
\sin 2x=2\sin x\cos x
\]
and
\[
\cos 2x=2\cos^2x-1
\]
Substituting these values:
\[
LHS=\frac{\sin x+2\sin x\cos x}{1+\cos x+2\cos^2x-1}
\]
\[
=\frac{\sin x(1+2\cos x)}{\cos x+2\cos^2x}
\]
Taking common factor \(\cos x\) from denominator:
\[
=\frac{\sin x(1+2\cos x)}{\cos x(1+2\cos x)}
\]
Cancel common terms:
\[
=\frac{\sin x}{\cos x}
\]
\[
=\tan x
\]
Hence proved,
\[
\boxed{\frac{\sin x+\sin 2x}{1+\cos x+\cos 2x}=\tan x}
\]